The machine (M) comprises a truncoconical rotating turret (3) having multiple radial heads (4), activated intermittently such that each of the radial heads (4) occupies, in succession, at least four consecutive stations (S1, S2, S3, S4), arranged in a circle about the turret (3). Loading organs (10) are located in the first station (S1), designed to locate a tray container (1) in the corresponding radial head 4, with an inclined disposition and with the upper head (1D) open. Supply organs (20) are comprised in the second station (S2) for progressively piling a batch of cards (2) in the corresponding container (1). Operating means (7) are associated to the turret (3), for closing the upper head (1D) of the container (1) in phase relation with a pause of the container (1) in the third station (S3). Extraction and orientation organs (80) are located in the fourth station (S4) for collecting the filled container (1) from the corresponding radial head (4) and arranging it horizontally on a plane provided by a side, without subjecting the container (1) to stress.